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Functional limit theorems for some self-similar Gaussian processes in critical and subcritical cases

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  • Liu, Heguang

Abstract

In this paper, under certain conditions, we investigate the asymptotic behavior of {∫0tfα(nH(Xs−λ))ds,t≥0}, where fα is the density of symmetric α-stable random variables with α∈(0,2) and X={Xt,t≥0} is some self-similar Gaussian process with index H∈(0,1). We mainly focus on the critical case H(2α+1)=1 and the subcritical case H(2α+1)<1. This work will extend the corresponding results in Hong et al. (2024) and may give another definition for the fractional derivative of local times of the Gaussian process X.

Suggested Citation

  • Liu, Heguang, 2026. "Functional limit theorems for some self-similar Gaussian processes in critical and subcritical cases," Statistics & Probability Letters, Elsevier, vol. 227(C).
    • Handle: RePEc:eee:stapro:v:227:y:2026:i:c:s0167715225001920
    DOI: 10.1016/j.spl.2025.110547
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    References listed on IDEAS

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    1. Nualart, David & Xu, Fangjun, 2014. "Central limit theorem for functionals of two independent fractional Brownian motions," Stochastic Processes and their Applications, Elsevier, vol. 124(11), pages 3782-3806.
    2. Song, Jian & Xu, Fangjun & Yu, Qian, 2019. "Limit theorems for functionals of two independent Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 129(11), pages 4791-4836.
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